metabelian, supersoluble, monomial, 3-hyperelementary
Aliases: C13⋊He3, C39.6C32, (C3×C39)⋊2C3, C32⋊(C13⋊C3), (C3×C13⋊C3)⋊C3, C3.6(C3×C13⋊C3), SmallGroup(351,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13⋊He3
G = < a,b,c,d | a13=b3=c3=d3=1, ab=ba, ac=ca, dad-1=a9, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C13⋊He3
►On 117 points
Generators in S117
(1 2 3 4 5 6 7 8 9 10 11 12 13)(14 15 16 17 18 19 20 21 22 23 24 25 26)(27 28 29 30 31 32 33 34 35 36 37 38 39)(40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)
(1 88 46)(2 89 47)(3 90 48)(4 91 49)(5 79 50)(6 80 51)(7 81 52)(8 82 40)(9 83 41)(10 84 42)(11 85 43)(12 86 44)(13 87 45)(14 99 61)(15 100 62)(16 101 63)(17 102 64)(18 103 65)(19 104 53)(20 92 54)(21 93 55)(22 94 56)(23 95 57)(24 96 58)(25 97 59)(26 98 60)(27 105 68)(28 106 69)(29 107 70)(30 108 71)(31 109 72)(32 110 73)(33 111 74)(34 112 75)(35 113 76)(36 114 77)(37 115 78)(38 116 66)(39 117 67)
(1 35 17)(2 36 18)(3 37 19)(4 38 20)(5 39 21)(6 27 22)(7 28 23)(8 29 24)(9 30 25)(10 31 26)(11 32 14)(12 33 15)(13 34 16)(40 70 58)(41 71 59)(42 72 60)(43 73 61)(44 74 62)(45 75 63)(46 76 64)(47 77 65)(48 78 53)(49 66 54)(50 67 55)(51 68 56)(52 69 57)(79 117 93)(80 105 94)(81 106 95)(82 107 96)(83 108 97)(84 109 98)(85 110 99)(86 111 100)(87 112 101)(88 113 102)(89 114 103)(90 115 104)(91 116 92)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(14 21 16)(15 24 25)(18 20 26)(19 23 22)(27 37 28)(29 30 33)(31 36 38)(32 39 34)(40 59 74)(41 62 70)(42 65 66)(43 55 75)(44 58 71)(45 61 67)(46 64 76)(47 54 72)(48 57 68)(49 60 77)(50 63 73)(51 53 69)(52 56 78)(79 112 99)(80 115 95)(81 105 104)(82 108 100)(83 111 96)(84 114 92)(85 117 101)(86 107 97)(87 110 93)(88 113 102)(89 116 98)(90 106 94)(91 109 103)
G:=sub<Sym(117)| (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,88,46)(2,89,47)(3,90,48)(4,91,49)(5,79,50)(6,80,51)(7,81,52)(8,82,40)(9,83,41)(10,84,42)(11,85,43)(12,86,44)(13,87,45)(14,99,61)(15,100,62)(16,101,63)(17,102,64)(18,103,65)(19,104,53)(20,92,54)(21,93,55)(22,94,56)(23,95,57)(24,96,58)(25,97,59)(26,98,60)(27,105,68)(28,106,69)(29,107,70)(30,108,71)(31,109,72)(32,110,73)(33,111,74)(34,112,75)(35,113,76)(36,114,77)(37,115,78)(38,116,66)(39,117,67), (1,35,17)(2,36,18)(3,37,19)(4,38,20)(5,39,21)(6,27,22)(7,28,23)(8,29,24)(9,30,25)(10,31,26)(11,32,14)(12,33,15)(13,34,16)(40,70,58)(41,71,59)(42,72,60)(43,73,61)(44,74,62)(45,75,63)(46,76,64)(47,77,65)(48,78,53)(49,66,54)(50,67,55)(51,68,56)(52,69,57)(79,117,93)(80,105,94)(81,106,95)(82,107,96)(83,108,97)(84,109,98)(85,110,99)(86,111,100)(87,112,101)(88,113,102)(89,114,103)(90,115,104)(91,116,92), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(14,21,16)(15,24,25)(18,20,26)(19,23,22)(27,37,28)(29,30,33)(31,36,38)(32,39,34)(40,59,74)(41,62,70)(42,65,66)(43,55,75)(44,58,71)(45,61,67)(46,64,76)(47,54,72)(48,57,68)(49,60,77)(50,63,73)(51,53,69)(52,56,78)(79,112,99)(80,115,95)(81,105,104)(82,108,100)(83,111,96)(84,114,92)(85,117,101)(86,107,97)(87,110,93)(88,113,102)(89,116,98)(90,106,94)(91,109,103)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,88,46)(2,89,47)(3,90,48)(4,91,49)(5,79,50)(6,80,51)(7,81,52)(8,82,40)(9,83,41)(10,84,42)(11,85,43)(12,86,44)(13,87,45)(14,99,61)(15,100,62)(16,101,63)(17,102,64)(18,103,65)(19,104,53)(20,92,54)(21,93,55)(22,94,56)(23,95,57)(24,96,58)(25,97,59)(26,98,60)(27,105,68)(28,106,69)(29,107,70)(30,108,71)(31,109,72)(32,110,73)(33,111,74)(34,112,75)(35,113,76)(36,114,77)(37,115,78)(38,116,66)(39,117,67), (1,35,17)(2,36,18)(3,37,19)(4,38,20)(5,39,21)(6,27,22)(7,28,23)(8,29,24)(9,30,25)(10,31,26)(11,32,14)(12,33,15)(13,34,16)(40,70,58)(41,71,59)(42,72,60)(43,73,61)(44,74,62)(45,75,63)(46,76,64)(47,77,65)(48,78,53)(49,66,54)(50,67,55)(51,68,56)(52,69,57)(79,117,93)(80,105,94)(81,106,95)(82,107,96)(83,108,97)(84,109,98)(85,110,99)(86,111,100)(87,112,101)(88,113,102)(89,114,103)(90,115,104)(91,116,92), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(14,21,16)(15,24,25)(18,20,26)(19,23,22)(27,37,28)(29,30,33)(31,36,38)(32,39,34)(40,59,74)(41,62,70)(42,65,66)(43,55,75)(44,58,71)(45,61,67)(46,64,76)(47,54,72)(48,57,68)(49,60,77)(50,63,73)(51,53,69)(52,56,78)(79,112,99)(80,115,95)(81,105,104)(82,108,100)(83,111,96)(84,114,92)(85,117,101)(86,107,97)(87,110,93)(88,113,102)(89,116,98)(90,106,94)(91,109,103) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13),(14,15,16,17,18,19,20,21,22,23,24,25,26),(27,28,29,30,31,32,33,34,35,36,37,38,39),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,88,46),(2,89,47),(3,90,48),(4,91,49),(5,79,50),(6,80,51),(7,81,52),(8,82,40),(9,83,41),(10,84,42),(11,85,43),(12,86,44),(13,87,45),(14,99,61),(15,100,62),(16,101,63),(17,102,64),(18,103,65),(19,104,53),(20,92,54),(21,93,55),(22,94,56),(23,95,57),(24,96,58),(25,97,59),(26,98,60),(27,105,68),(28,106,69),(29,107,70),(30,108,71),(31,109,72),(32,110,73),(33,111,74),(34,112,75),(35,113,76),(36,114,77),(37,115,78),(38,116,66),(39,117,67)], [(1,35,17),(2,36,18),(3,37,19),(4,38,20),(5,39,21),(6,27,22),(7,28,23),(8,29,24),(9,30,25),(10,31,26),(11,32,14),(12,33,15),(13,34,16),(40,70,58),(41,71,59),(42,72,60),(43,73,61),(44,74,62),(45,75,63),(46,76,64),(47,77,65),(48,78,53),(49,66,54),(50,67,55),(51,68,56),(52,69,57),(79,117,93),(80,105,94),(81,106,95),(82,107,96),(83,108,97),(84,109,98),(85,110,99),(86,111,100),(87,112,101),(88,113,102),(89,114,103),(90,115,104),(91,116,92)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(14,21,16),(15,24,25),(18,20,26),(19,23,22),(27,37,28),(29,30,33),(31,36,38),(32,39,34),(40,59,74),(41,62,70),(42,65,66),(43,55,75),(44,58,71),(45,61,67),(46,64,76),(47,54,72),(48,57,68),(49,60,77),(50,63,73),(51,53,69),(52,56,78),(79,112,99),(80,115,95),(81,105,104),(82,108,100),(83,111,96),(84,114,92),(85,117,101),(86,107,97),(87,110,93),(88,113,102),(89,116,98),(90,106,94),(91,109,103)]])
47 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 13A | 13B | 13C | 13D | 39A | ··· | 39AF |
| order | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 13 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 1 | 1 | 3 | 3 | 39 | ··· | 39 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
47 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | He3 | C13⋊C3 | C3×C13⋊C3 | C13⋊He3 |
| kernel | C13⋊He3 | C3×C13⋊C3 | C3×C39 | C13 | C32 | C3 | C1 |
| # reps | 1 | 6 | 2 | 2 | 4 | 8 | 24 |
Matrix representation of C13⋊He3 ►in GL3(𝔽79) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 10 | 25 |
| 55 | 8 | 57 |
| 57 | 72 | 11 |
| 11 | 9 | 31 |
| 23 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 23 |
| 1 | 0 | 0 |
| 24 | 68 | 54 |
| 3 | 55 | 10 |
G:=sub<GL(3,GF(79))| [0,0,1,1,0,10,0,1,25],[55,57,11,8,72,9,57,11,31],[23,0,0,0,23,0,0,0,23],[1,24,3,0,68,55,0,54,10] >;
C13⋊He3 in GAP, Magma, Sage, TeX
C_{13}\rtimes {\rm He}_3 % in TeX
G:=Group("C13:He3"); // GroupNames label
G:=SmallGroup(351,8);
// by ID
G=gap.SmallGroup(351,8);
# by ID
G:=PCGroup([4,-3,-3,-3,-13,97,1299]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
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